Let us generate every subset of \(n\)  of size exactly \(k\) . There are \(n^k\)  such subsets. For each subset \(U\) , let us create a set of forced edges \(\lbrace  \lbrace a_i, \operatorname{first}(a_i) \rbrace  | a_i \in U \rbrace \) , where \(\operatorname{first}(a_i)\)  is the agent \(a_j\)  such that \(\operatorname{rank}(a_i, a_j) = 1\) . We can solve in \(O(m)\)  time, where \(m\)  is the number of all the acceptable pairs, whether there is a stable matching with the forced edges present [1]}. If such a matching exists, we can report this as a solution. If no such matching exists for any subset \(U\) , we do not have stable matching with at least \(k\)  first choices.
